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Zbl 0686.08008
Willard, Ross
Congruence lattices of powers of an algebra. (English)
[J] Algebra Univers. 26, No.3, 332-340 (1989). ISSN 0002-5240; ISSN 1420-8911/e

Let $\Cal P$ be a property which can be attributed to 0-1 sublattices of the equivalence lattice on an arbitrary set. It was already proved by S. Burris and the author that for every integer $k\ge 2$ there is an $n\ge 1$ such that for every algebra $A$ of cardinality $k$, if $\text{Con}(A\sp m)$ satisfies $\Cal P$ for all $m\le n$ then $\text{Con}(A\sp m)$ satisfies $P$ for all $m$. Let $n\sb{\Cal P}(k)$ denote the least such $n$. The paper evaluates the function $n\sb{\Cal P}$ for $p=$ permutability, distributivity, arithmeticity, weak distributivity, modularity, and the property of being skew-free (= the Fraser-Horn property). Main results: (i) $n\sb{\text{Perm}}(k)\le k\sp 3$; (ii) $n\sb{\text{Dist}}(k)\le k\sp{k+1}$; (iii) $n\sb{\text{Skew-free}}(k)\le k\sp3 + k\sp2 - k$; (iv) $n\sb{\text{Mod}}(k)\le k\sp{4m(k)}$, where $m(k)=k\sp{(k\sp 4-k\sp3 + k\sp2)} - 1$.
[J.Duda]

MSC 2000:: *08B10 Congruence modularity and generalizations in varieties of algebras
08A30 Subalgebras of general algebraic systems

Keywords: power of algebra; congruence lattice; permutability; distributivity; arithmeticity; weak distributivity; modularity; skew-free; Fraser-Horn property

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